Ejecución de la auditoría al ciclo de inventarios.

ˆ The reduction coincides with a known one, therefore we can link the solubilities of the two cases by linear equations.

In most cases, all three outcomes are possible for the same initial reduction type, depending on the coecients of the quartics. At each stage we split the computation in subcases, imposing congruence conditions on the coecients. We use the notation

c(XiYjZk)for the coecient of the monomialXiYjZk. Since most of the computations

involve the valuations of the coecients of the quarticsT overZp we represent them in a

triangle, where for each point the distance from each vertex determines the exponent of each variable and the value is the valuations of the coecient of the associated monomial.

Z4 · · · · · · · · · · X4 · · · Y4 i k j

where each dot refers tov(c(XiYjZk)), i.e. the valuation of the coecient of the mono-

mial XiYjZk, for one of the 15 triples (i, j, k). For instance if the reduction T is a

quadruple line which, after a suitable change of coordinates, we may assume is X4 = 0.

In this case, from the reduction we have information only on the coecient ofX4, which

has valuation 0, while all the other coecients have valuation≥1, therefore the triangle

of valuations would look like this:

Z4 ≥1

≥1 ≥1

≥1 ≥1 ≥1

≥1 ≥1 ≥1 ≥1

X4 = 0 ≥1 ≥1 ≥1 ≥1 Y4

7.2 Probability tools and further counts

Throughout the detailed computation of the probabilities of each case, we will need some general results; we list most of them here in order to make the exposition clearer.

7.2. Probability tools and further counts

Sometimes the reduction of the quartic overFp is a double line times a quadratic

whose coecients are parametrized by polynomials in one variable; for instance this happens in the proof of Proposition 7.3.13. The solubility of those congurations is related to the factorisation of the quadratic, which is described for oddpby the Legendre

symbol of its discriminant.

We consider a parametrized quadratic M(b, X) = h(b)X2 +g(b)X+ 1, where h, g ∈ Fp[b]. For each b ∈ Fp, M(b, X) has either distinct roots in Fp, a double one in

Fp or two conjugate ones inFp2, according to the value of the Legendre symbol

_∆(b)

p

where∆(b) =g2−4h.

Therefore, dening the setS asS :=

_∆(b) p :b∈_Fp , we say thatM is

(i) Good, if 1 ∈ S, i.e. there exists a b such that the polynomial M(b, X) has two

distinct roots on Fp.

(ii) Bad, if S={−1}, i.e. for anyb∈_Fp M(b, x)has two conjugate roots on Fp2.

(iii) Undetermined, if {0} ⊆S ⊆ {0,−1}, i.e. there are no b such that the polynomial M(b, X)has two distinct roots onFp but there exists one for whichM has a double

root.

Lemma 7.2.1. Let p > 17 be a prime and h(b) and g(b) be two polynomials in Fp[b],

whose degrees are bounded respectively by 4 and 3.

Let u be the leading coecient of ∆(b), the discriminant of M(b, X) =h(b)X2+ g(b)X+ 1, whereM is the polynomial described above. Then, we have the following count

for all the possible cases forT: deg(∆)

u p

Good Undetermined Bad 01 _±1 p−1 2 1 p−1 2 1 ±1 p2−p 0 0 2 1 p−1 2 p2 0 0 2 ±1 2p3−₂3p2+p p2₂−p 0 3 ±1 (p−1)p3 0 0 4 1 p−1 2 p 4 _{0 0} 4 ±1 2p5−2p4₂−p3+p2 (p−1)(₄p2+p) (p−1)(₄p2−p) 6 1 p−1 2 p 6 _{0 0}

1_{We are including in the case ofdeg(∆) = 0the null polynomial∆ = 0, which is undetermined.}

7.2. Probability tools and further counts

Proof. First of all, we refer to an argument of M.Bhargava, J. Cremona, and T. Fisher from [BCF]. They proved that whenp≥deg ∆2 if ∆(b) is not of the formu0t2(b), with u0 non-quadratic residue and t ∈ _Fp[b], then M is good. Their proof, adapted to our

setting, is the following. According asdeg(∆)is odd or even we writedeg(∆) = 2i+ 1or 2i+ 2. We write∆ = ∆1t2 where∆1 andt∈Fp[b], with ∆1 squarefree. Ifdeg(∆1)≥1,

then the Hasse-Weil bound gives a lower bound of (p−1−2i√p)/2 for the number of

elements b inFp such that ∆1(b) is a square. Since the number of roots of tis at most i, ifp > (2i+ 1)2, the polynomial M is good. If ∆1 =c is a constant, then in the case

c p

= +1the polynomial M is good if there exists an element of Fp that is not a root

oft, which exists when p > g+ 1.

To complete the proof of this Lemma, it remains to discuss, for even degrees, when∆(b) =u0t2(b), with u0 a non-quadratic residue. These cases are the undetermined

and bad ones.

If∆(b) is the null polynomial then M is an undetermined case.

When the degree of∆is zero we have(p−1)/2 choices foru0 that correspond to (p−1)/2 bad polynomials.

When the degree of ∆ is two, t is a linear polynomial, hence it has a root and

thereforeM is undetermined. We have pchoices for tand (p−1)/2 for u0.

When the degree of ∆is four, tcan be either irreducible overFp, in such caseM

is bad, or having at least one root, in this caseM is undetermined. In the rst case there

are(p2−p)/2possiblet, in the second (p2+p)/2; times (p−1)/2 choices foru0.

To conclude, it is possible to obtain a sharper lower bound for p, instead of p >deg(∆)2. Indeed, by an exhaustive method for the polynomials dened over small

prime elds, we worked out thatp >17 is sucient.

Another general result we will use during the computation of the probabilities is related to the distribution of the values of a polynomial at a xed set of elements.

In the following Theorem we consider polynomialsf(x)∈_Fp[x]whose coecients

are random Fp-valued variables.

Theorem 7.2.2. Let B = {b1, . . . , bk} ⊂ Fp be a set of k distinct elements in Fp.

Let f(x) ∈ Fp[x] be a random polynomial whose rst k coecients are uniformly dis-

tributed over Fkp and are independent of the remaining coecients. Then the values

f(b1), . . . , f(bk) are also uniformly distributed

Proof. LetP be thek×(n+ 1)matrix (bj_i), letF be the vector column of the coeents fi of f(x) and let V be the vector column of the k values f(bi) of the polynomial f at